Congruence on a group

This article is about a basic definition in group theory. The article text may, however, contain advanced material.
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Definition

Symbol-free definition

A congruence on a group is an equivalence relation on the elements of the group that is compatible with all the group operations.

Definition with symbols

A congruence on a group $G$ is an equivalence relation $\equiv$ on $G$ such that:

• $a\equiv b\implies a^{-1}\equiv b^{-1}$
• $a\equiv b,c\equiv d\implies ac\equiv bd$

The term congruence can more generally be used for any algebra, in the theory of universal algebras. Further information: congruence on an algebra

Facts

The congruence class of the identity element

It is easy to see that the congruence class of the identity element is a normal subgroup.

Conversely, given any normal subgroup, there is a unique congruence where the congruence class of the identity element is that normal subgroup. The congruence classes here are the cosets of the normal subgroup.

The quotient map for a congruence

Given a congruence on a group, there is a natural quotient map from the group to the set of congruence classes. Under this map, the set of congruence classes inherits a group structure. This is termed the quotient group.